Theorem sbf 870

Description: Substitution for a variable not free in a wff does not affect it.

Hypothesis

Ref	Expression
sbf.1	|- (ph -> A.xph)

Assertion

Ref	Expression
sbf	|- ([y / x]ph <-> ph)

Proof of Theorem sbf

Step	Hyp	Ref	Expression
1		sb1 858	. . . 4 |- ([y / x]ph -> E.x(x = y /\ ph))
2		sbf.1	. . . . 5 |- (ph -> A.xph)
3	2	19.41 774	. . . 4 |- (E.x(x = y /\ ph) <-> (E.x x = y /\ ph))
4	1, 3	sylib 173	. . 3 |- ([y / x]ph -> (E.x x = y /\ ph))
5	4	pm3.27d 262	. 2 |- ([y / x]ph -> ph)
6		stdpc4 869	. . 3 |- (A.xph -> [y / x]ph)
7	2, 6	syl 12	. 2 |- (ph -> [y / x]ph)
8	5, 7	impbi 139	1 |- ([y / x]ph <-> ph)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797  [wsb 852

This theorem is referenced by:  sb6x 871  sb19.21 888  sbrbif 893  sbequ5 898  sbid2 911  sb5f1 917  sbabel 1189  sbcgf 1469

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853

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